­­­HPC implementation and algorithm of the Thermal Lattice Boltzmann Method for geodynamics simulations in 2D and 3D

The Thermal Lattice Boltzmann Method (TLBM) is a powerful numerical method for thermally driven fluid flow simulations that is starting to be applied to geodynamics research. It is based on solving the Boltzmann equations on a discrete lattice and involves two steps of movement and collision of particle number densities carrying mass density and energy density on a discrete lattice. The collision step is achieved by relaxing the distributions to the equilibrium distribution where the relaxation times relate to the kinematic viscosity and thermal diffusivity of the fluid. We present the TLBM algorithm and an optimized HPC implementation of the TLBM where the main code is written in python using MPI for python, and this code calls highly optimized c functions for the kernels which do the heavy computational work. The same code works in 2D or 3D and we calculate the optimal 2D or 3D domain decomposition at the start of each run. Edges of domains are sent and received using optimal unblocking MPI requests, with the send and receive requests and buffers initialized at the start of a run to further optimize the communication costs. We present performance results which show near linear speedup to thousands of cores provided the domain size is not too small. We achieve of order 2-3 Gflops per core which is typically over 50% of peak performance. We show 2D runs using a highly nonlinear rheology which promotes the formation of plate-tectonic like dynamics with upwelling and downwelling plumes with the horizontal motion tending to be constrained to the upper 100km of the model. We also show 2D and 3D runs with temperature dependent viscosities and power law thermal boundary layer scaling with Nusselt number. And we show runs of simulations with high Rayleigh numbers up to 10**12 and Prandtl numbers up to 10**4. The TLBM offers a means to study the effect of highly nonlinear rheologies on geodynamical processes, and may eventually lead to a more complete simulation capability for studies of planet and exoplanet evolution.